Multiple eigenvalues for the Steklov problem in a domain with a small hole. A functional analytic approach

Abstract

Let $\mathit{\alpha }\in ]0\mathrm{,}1[$ . Let ${\mathrm{\Omega }}^{\mathit{o}}$ be a bounded open domain of ${\mathbb{R}}^{\mathit{n}}$ of class ${\mathit{C}}^{1\mathrm{,}\mathit{\alpha }}$ . Let ${\mathit{\nu }}_{{\mathrm{\Omega }}^{\mathit{o}}}$ denote the outward unit normal to $\partial {\mathrm{\Omega }}^{\mathit{o}}$ . We assume that the Steklov problem $\mathrm{\Delta }\mathit{u}=0$ in ${\mathrm{\Omega }}^{\mathit{o}}$ , $\frac{\partial \mathit{u}}{\partial {\mathit{\nu }}_{{\mathrm{\Omega }}^{\mathit{o}}}}=\mathit{\lambda }\mathit{u}$ on $\partial {\mathrm{\Omega }}^{\mathit{o}}$ has a multiple eigenvalue $\tilde{\mathit{\lambda }}$ of multiplicity r. Then we consider an annular domain $\mathrm{\Omega }\mathrm{(}\mathit{ϵ}\mathrm{)}$ obtained by removing from ${\mathrm{\Omega }}^{\mathit{o}}$ a small cavity of class ${\mathit{C}}^{1\mathrm{,}\mathit{\alpha }}$ and size $\mathit{ϵ}\mathrm{>}0$ , and we show that under appropriate assumptions each elementary symmetric function of r eigenvalues of the Steklov problem $\mathrm{\Delta }\mathit{u}=0$ in $\mathrm{\Omega }\mathrm{(}\mathit{ϵ}\mathrm{)}$ , $\frac{\partial \mathit{u}}{\partial {\mathit{\nu }}_{\mathrm{\Omega }\mathrm{(}\mathit{ϵ}\mathrm{)}}}=\mathit{\lambda }\mathit{u}$ on $\partial \mathrm{\Omega }\mathrm{(}\mathit{ϵ}\mathrm{)}$ which converge to $\tilde{\mathit{\lambda }}$ as ϵ tend to zero, equals real a analytic function defined in an open neighborhood of $\mathrm{(}0\mathrm{,}0\mathrm{)}$ in ${\mathbb{R}}^2$ and computed at the point $\mathrm{(}\mathit{ϵ}\mathrm{,}{\mathit{\delta }}_{2\mathrm{,}\mathit{n}}\mathit{ϵ}log\mathit{ϵ}\mathrm{)}$ for $\mathit{ϵ}\mathrm{>}0$ small enough. Here ${\mathit{\nu }}_{\mathrm{\Omega }\mathrm{(}\mathit{ϵ}\mathrm{)}}$ denotes the outward unit normal to $\partial \mathrm{\Omega }\mathrm{(}\mathit{ϵ}\mathrm{)}$ , and ${\mathit{\delta }}_{2\mathrm{,}2}\equiv 1$ and ${\mathit{\delta }}_{2\mathrm{,}\mathit{n}}\equiv 0$ if $\mathit{n}⩾3$ . Such a result is an extension to multiple eigenvalues of a previous result obtained for simple eigenvalues in collaboration with S. Gryshchuk.